Oldroyd-B constitutive equation

A frequently used example of Oldroyd-type constitutive equations is the Oldroyd-B model. The Oldroyd-B model can be thought of as a description of the constitutive behaviour of a fluid made by the dissolution of a (UCM) fluid in a Newtonian solvent . Here, the parameter A, called the retardation time is de.fined as A = A (r s/(ri + s), where 7]s is the viscosity of the solvent. Hence the extra stress tensor in the Oldroyd-B model is made up of Maxwell and solvent contributions. The Oldroyd-B constitutive equation is written as... 

Schmidt et al. (102) carried out a detailed experimental study of PET blow molding with a well-instrumented machine and compared the results with theoretical predictions using FEM and an Oldroyd B constitutive equation. They measured and calculated internal gas pressure, coupled it with the thermomechanical inflation and performed experiments and computations with free parison inflation. 

Here r is given by the UCM equation 4.3.7 (or equivalently by the Lodge equation, eq.4.3.18), and is usually just a Newtonian term 2r7jD, where rj, is the solvent viscosity. The combination of these two terms is the Oldroyd-B constitutive equation (Oldroyd, 1950 see Exercise 4.6.4). Figure 4.3.4 compares the storage mod-... 

The three constant Oldroyd model is a nonlinear constitutive equation of the differential corrotational type, such as the Zaremba-Fromm-Dewitt (ZFD) fluid (Eq. 3.3-11). [For details, see R. B. Bird, R. C. Armstrong, and O. Hassager, Dynamics of Polymeric Liquids, Second Edition, Vol. 1, Wiley, New York, 1987, Table 7.3-2.]... 

The constitutive equations are the Oldroyd-B model and a modified Oldroyd-B model in which the viscosity depends on the rate of strain. In [79], Laure et al. study the spectral stability of the plane Poiseuille flow of two viscoelastic fluids obeying an Oldroyd-B law in two configurations the first one is the two layer Poiseuille flow in the second case the same fluid occupies the symmetric upper and lower layers, surrounding the central fluid. (See Figure 9.)... 

In the story of numerical flow simulation, the ability to predict observed and significant viscoelastic flow phenomena of polymer melts and solutions in an abrupt contraction has been unsuccessful for many years, in relation to the incomplete rheological characterization of materials, especially in elongation. The numerical treatments have often been confined to flow of elastic fluids with constant viscosity, described by differential constitutive equations as the Upper Convected Maxwell and Oldroyd-B models. Fortunately, the recent possibility to use real elastic fluids with constant viscosity, the so-called Boger fluids [10], has narrowed the gap between experimental observation and numerical prediction [11]. 

It should be pointed out that the improvement of convergence might also be related to realistic preditions of shear and elongational viscosities by the Phan-Thien Tanner model, when compared to the Upper Convected Maxwell, Oldroyd-B and White-Metzner models. Satisfactory munerical results were also obtained with multi-mode integral constitutive equations using a spectnun of relaxation times [7, 17, 20-27], such as the K-BKZ model in the form introduced by Papanastasiou et al. [19]. 

The streamline patterns are quite identical for both constitutive equations. However, the vortex is more pronounced for the multimode Phan-Thien Tanner model, whereas the swelling is greater for the generalized Oldroyd-B model... 

Equations (3-32)-(3-34) are equivalent to the so-called Oldroyd-B equation. The Oldroyd-B equation is a simple, but qualitatively useful, constitutive equation for dilute solutions of macromolecules (see Section 3.6.2). Refinements to the simple elastic dumbbell model, such as the effects of the nonlinearity of the force-extension relationship at high extensions, are discussed in Section 3.6.2.2.I. 

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